Singular Yamabe problem for scalar flat metrics on the sphere

Let \(\Omega \) be a domain on the unit n-sphere \( {\mathbb {S}}^n\) and \( \overset{{\,}_\circ }{g}\) the standard metric of \({\mathbb {S}}^n\), \(n\ge 3\). We show that there exists a conformal metric g with vanishing scalar curvature \(R(g)=0\) such that \((\Omega , g)\) is complete if and only if the Bessel capacity \({\mathcal {C}}_{\alpha , q}({\mathbb {S}}^n\setminus \Omega )=0\), where \(\alpha =1+\frac{2}{n}\) and \(q=\frac{n}{2}\). Our analysis utilizes some well known properties of capacity and Wolff potentials, as well as a version of the Hopf–Rinow theorem for the divergent curves.